Department of Applied Mathematics HAIT – Holon Academic Institute of Technology Holon, ISRAEL...

Department of Applied Mathematics

HAIT – Holon Academic Institute of TechnologyHolon, ISRAEL

Workshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004

Towards exact integrabilityof replica field theories

H A I T

Eugene Kanzieper

Based on:Replica field theories, Painlevé transcendents,

and exact correlation functions, Phys. Rev. Lett. 89, 250201 (2002)

Thanks to:Craig Tracy (UCal-Davis), Peter Forrester (UMelb)

(for guiding through Painlevé literature)

Discussions with: Alex Kamenev (UMin), Ady Stern (WIS), Jac Verbaarschot

Supported by:Albert Einstein Minerva Centre for Theoretical

Physics(Weizmann Institute of Science, Rehovot, Israel)

(SU

NY)

Outline

Nonperturbative Methods in Physics of Disorder

Replica sigma models

Supersymmetric sigma model

Keldysh sigma model

Why Replicas? What Are Replicas and The Replica Limit?

Two Pitfalls: Analytic Continuation and (Un) controlled Approximations

Message: exact approach to replicas needed

Outline

Nonperturbative Methods in Physics of Disorder

Towards Exact Integrability of Replica Field

Theories in 0 Dimensions

Conclusions

Why Replicas? What Are Replicas and The Replica Limit?

Two Pitfalls: Analytic Continuation and (Un) controlled Approximations

Outline

Nonperturbative Methods in Physics of Disorder

Outline

Nonperturbative Methods in Physics of Disorder

Statistical description: Ensemble of grains

Ensemble averaged observable

Disorder as a perturbation ?..

11 FE

Disordered grain

Qu

an

tum

dot

mean scattering time

Fermi energy

IMPORTANTPHYSICS

LOST

Whole issue of strong localisation

Disordered grain

Qu

an

tum

dot

Weak disorder limit: long time particle evolution (times larger than the Heisenberg time)

11 FE

Disordered grain

Qu

an

tum

dot

Whole issue of strong localisation

F a i l u r e

!

to perturbative treatment of disorder

IMPORTANTPHYSICS

LOST

Disordered grain

Qu

an

tum

dot

NO

Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982

Supersymmetric sigma modelEfetov 1982

Keldysh sigma modelHorbach, Schön 1990 Kamenev, Andreev 1999

Q

Field Theoretic Approaches

finite dimensionalmatrix field

nn Q

symmetry

QQQ JiDdF 0J 2Tr

8)( 2r

Q DQZ )(exp JFJ

1Q 2

A Typical NonlinearNonlinearly constrained matrix field of

certain symmetries

Generating functional

Action

Model

Outline

Nonperturbative Methods in Physics of Disorder

Replica sigma models

Supersymmetric sigma model

Keldysh sigma model

Perturbative approach may become quite useless even in the weak disorder limit Nonperturbative approaches: the three formulations

Why Replicas? What Are Replicas and the

Replica Limit?

Outline

Nonperturbative Methods in Physics of Disorder

Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982

Supersymmetric sigma modelEfetov 1982

Keldysh sigma modelHorbach, Schön 1990 Kamenev, Andreev 1999

Field Theoretic Approaches

disorder

interaction

interactionout-of-equilibrium

disorderdisorder

Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982

Field Theoretic Approaches

interactiondisorder

A viable tool to treat an interplay between disorder

and interaction!

Replica sigma models (bosonic and fermionic)Wegner 1979Larkin, Efetov, Khmelnitskii 1980Finkelstein 1982

Field Theoretic Approaches

interactiondisorder

A viable tool to treat an interplay between disorder

and interaction!

Why Replicas? What Are Replicas and the

Replica Limit?

Outline

Nonperturbative Methods in Physics of Disorder

Replica trick

Mean level density out of

Looks easier (replica partition function)

Reconstruct through the replica limit

based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

HE

EGˆ

1Tr)(

)ˆ(det)( HE EZ nn

)(1

)( lim0

EZE

EG nn n

commutativity!

Replica trick

Density-density correlation function out of

Looks easier (replica partition function)

Reconstruct through the replica limit

based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

commutativity!

HE

HE EG EG

ˆ1

Trˆ

1Tr)()(

21

21

)ˆ(det)ˆ(det),( 2121 HE HE E EZ nnn

),(1

)()( 2121

2

20

21 lim E EZE E

EG EG nn n

Reconstruct through the replica limit

based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

commutativity!),(1

)()( 2121

2

20

21 lim E EZE E

EG EG nn n

Replica trick

Word of caution:

for more than two decades no one could rigorously implement it in

mesoscopics !

Replica Trick: A Bit of Chronology

1979 1980 1985

Bosonic Replicas

FermionicReplicas

F. Wegner

Larkin, Efetov, Khmelnitskii

First Critique(RMT)

Verbaarschot, Zirnbauer

Reconstruct through the replica limit

based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

commutativity!),(1

)()( 2121

2

20

21 lim E EZE E

EG EG nn n

Replica Trick: A Bit of Chronology

Reconstruct through the replica limit

based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

commutativity!),(1

)()( 2121

2

20

21 lim E EZE E

EG EG nn n

Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985

“… the replica trick … suffers from a serious drawback: it is mathematically ill founded.”

“… the replica trick for disordered electron systems is limited to those regions of parameter space where the nonlinear sigma model can be evaluated perturbatively.”

Another critique of the replica trick, M. Zirnbauer 1999

Replica Trick: A Bit of Chronology

1979 1980 1985

Bosonic Replicas

FermionicReplicas

F. Wegner

Larkin, Efetov, Khmelnitskii

First Critique(RMT)

Verbaarschot, Zirnbauer

Reconstruct through the replica limit

based on Edwards, Anderson 1975; Hardy, Littlewood, Pólya 1934

commutativity!),(1

)()( 2121

2

20

21 lim E EZE E

EG EG nn n

1999

Kamenev, Mézard

Replica SymmetryBreaking (RMT)

Second Critique(RMT)

Zirnbauer

Asymptotically Nonperturbative

Results:

?

19821983

SUSY

Efetov

Two Pitfalls: Analytic Continuation and (Un) controlled Approximations

Why Replicas? What Are Replicas and the

Replica Limit?

Outline

Nonperturbative Methods in Physics of Disorder

g0D limit:

Disordered grain

Qu

an

tum

dot

5.73 4.28 0.18 9.33 4.58 9.27 7.30 4.03 4.05 1.59 6.49 9.19

4.78 8.45 0.02 9.52 6.97 4.20 1.14 9.93 5.94 6.49 5.03 4.50

6.41 4.02 0.01 5.17 9.32 4.73 3.00 3.19 0.74 8.03 4.38 1.30

1.83 2.47 8.03 6.60 4.34 9.47 9.93 5.94 6.49 4.78 4.85 3.28

4.06 7.37 9.03 8.05 4.51 3.95 4.00 3.05 3.58 7.10 4.48 9.37

4.78 8.45 0.02 9.52 6.97 4.20 8.03 7.94 5.29 1.18 4.38 3.01

0.09 5.32 3.86 8.22 0.36 0.88 0.28 2.40 1.39 6.60 4.34 9.47

1.18 2.87 1.14 9.93 5.94 6.49 4.78 8.45 0.02 9.52 6.97 4.20

3.00 5.29 3.57 5.29 8.83 7.17 2.40 1.39 5.73 4.28 0.18 9.33

0.28 2.40 1.39 5.73 6.41 4.02 0.01 5.17 5.07 7.35 4.78 8.45

8.45 7.30 4.03 4.05 1.59 6.49 9.19 3.02 4.39 4.04 9.03 8.10

9.93 5.94 6.49 4.78 4.85 3.28 7.24 8.04 0.39 1.83 2.47 8.03

RMT

Two Pitfalls

(a) Analytic Continuation

)(1

ˆ1

Tr)( lim0

EZE

HE

EG nn n

)ˆ(det)( HE EZ nn

HEi DHE )ˆ(exp)ˆ(det

n

kkkkn

1

)ˆ(exp)(~ HEi D EZ

Original recipe

Field theoretic realisation

Rn

Zn

(a) Analytic Continuation

)ˆ(det)( HE EZ nn

n

kkkkn

1

)ˆ(exp)(~ HEi D EZ

Rn

Zn

van Hemmen and Palmer 1979Verbaarschot and Zirnbauer 1985Zirnbauer 1999

U n i q u e n e s

s

?..

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Saddle point evaluation for matrices of large dimensions

Rn,)ˆ(det)( HE EZ nn

Zn

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

known explicitlyReplica Symmetry Breaking for “causal” saddle points

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Saddle point evaluation for matrices of large dimensions

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

known explicitly)(~ Nn Obreaks down at

p

jpnpn jn

jG

npn

nG

1,, )1(

)()(vol,

)()(

)(

UUU

Analytic continuation…

0)(vol 1, npnG

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Saddle point evaluation for matrices of large dimensions

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

)(~ Nn Obreaks down at

Does NOT existIs NOT unique: Re-enumerate Saddles!!

Analytic continuation…

In the vicinity n=0:

Kamenev, Mézard 1999

Zirnbauer 1999Kanzieper

2004 (unpublished

)

diverges for Zn

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Saddle point evaluation for matrices of large dimensions

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

)(~ Nn Obreaks down at

Analytic continuation…..?

diverges for Zn

What’s the reason(s) for the failure ?

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Saddle point evaluation for matrices of large dimensions

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

)(~ Nn Obreaks down at

Analytic continuation…..?

diverges for Zn

1D 2D

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Saddle point evaluation for matrices of large dimensions

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

)(~ Nn Obreaks down at

Analytic continuation…..?

diverges for Zn

(b) (Un) Controlled Approximations

made prior to analytic continuation

DoS in GUEN from fermionic replicas a-lá Kamenev-Mézard (1999)

Saddle point evaluation for matrices of large dimensions

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

)(~ Nn Obreaks down at

Analytic continuation…..?

diverges for Zn

It’s a bit too dangerous to make analyticcontinuation based on an approximate result !

Two Pitfalls: Analytic Continuation and (Un) controlled Approximations

Why Replicas? What Are Replicas and the

Replica Limit?

Outline

Nonperturbative Methods in Physics of Disorder

Towards Exact Integrability of Replica Field

Theories in 0 Dimensions

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

),()(vol1~,0 ,

(sp) NzGN pnp pnnE E,Z

n Z

Saddle point evaluation for large matrices (KM, 1999)

duality

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Rn,)ˆ(det)(ˆ

N

nn GUE

H

HEEZ

Zn

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

Let’s do everything exactly !!

duality

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Rn,)ˆ(det)(ˆ

N

nn GUE

H

HEEZ

Zn

“There is of course little hope that the multi-dimensional integrals appearing in the replica formalism … can ever be evaluated non-perturbatively … ”Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

Let’s do everything exactly !!

duality

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Rn,)ˆ(det)(ˆ

N

nn GUE

H

HEEZ

Zn

Exact evaluation is possible as there exists an exact link between 0D replica field theories and the theory of nonlinear integrable hierarchies. EK: PRL 89, 250201 (2002)

“There is of course little hope that the multi-dimensional integrals appearing in the replica formalism … can ever be evaluated non-perturbatively … ”Critique of the replica trick, J. Verbaarschot and M. Zirnbauer 1985

Paul Painlevé(1863-1933)

Gaston Darboux (1842-1917)

No PhotoYet

Morikazu Toda

born 1917

French Prime Minister

September-November 1917

April-November 1925

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

Let’s do everything exactly !!

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Rn,)ˆ(det)(ˆ

N

nn GUE

H

HEEZ

Zn

n

kk

Nkn

n

kkn d

1

22

1

exp)(~ E i EZ

1det j

knn

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

Let’s do everything exactly !!

iE kk

112

1

22

)(~

j

kj,kj

kj,kiN

k

n

kk

nn

kkede det det EEEZ

2222

)(~ kjNij,k

nn ede EEEZ det

n

kk

Nkn

n

kkn d

1

22

1

exp)(~ E i EZ

1det j

knn

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

iE kk

112

1

22

)(~

j

kj,kj

kj,kiN

k

n

kk

nn

kkede det det EEEZ

2222

)(~ kjNij,k

nn ede EEEZ det

nn

Nikj

kjn

n ede

det EE

EEZ 2

,

22

)(~

n

kk

Nkn

n

kkn d

1

22

1

exp)(~ E i EZ

1det j

knn

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

iE kk

112

1

22

)(~

j

kj,kj

kj,kiN

k

n

kk

nn

kkede det det EEEZ

2222

)(~ kjNij,k

nn ede EEEZ det

nn

Nikj

kjn

n ede

det EE

EEZ 2

,

22

)(~

n

kk

Nkn

n

kkn d

1

22

1

exp)(~ E i EZ

1det j

knn

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

nn

Nikj

kjn

n ede

det EE

EEZ 2

,

22

)(~

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

Hankel determinant !!Darboux Theorem !!

),(~ Nn E

1),(~0

NE

),(~1 NE

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

nnkj

kjn

nn

n NeNe det ),(~),(~)(~

1,

22

EEEZ EEE

Q QE i

DQEZ ,Trexp)(det

)(~

2

)(

N

nnn

(Positive) Toda Lattice Equation

112 ~~~~~

nnnnn '''

Toda Lattice Hierarchy for replica partition functions

is a fingerprint of exact integrability hidden in replica field theories !

S Y M M

E T R Y

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

)(~),(~,1),(~110

2

EZE E E eNN

),(~)(exp)(~ 2 Nn nn E E EZ

112 ~~~~~

nnnnn '''

S Y M M

E T R Y

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

),(~)(exp)(~ 2 Nn nn E E EZ

112 ~~~~~

nnnnn '''

Does it help us perform analytic continuation

Zn Rnfrom to ?

Zn

Kyoto School’s Formalism

Okamoto, Noumi, Yamada

discovered a link between Toda Lattices and Painlevé equations

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Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

Toda Lattice can be reduced to Painlevé equationPainlevé equation contains the replica index as a parameter in its coefficients

A “simple minded” analytic continuation of so- obtained replica partition function away from integers leads to a correct replica limit

Loosely speaking:

Correctness of a such an analytic continuation can independently be proven (details in the papers), but uniqueness …

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

112 ~~~~~

nnnnn '''

),(~)(exp)(~ 2 Nn nn E E EZ Zn

E i

ZEZ0

IV )(exp)0(~)(~ tdtnn

02244 22 N'n'''t'' IVIVIVIVIVIV

+ boundary conditions

Rn

How it looks in the very end…

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

)(~1)( lim

0

EZE

EG nn n

get exact answer !

How it looks in the very end…

Exact Integrability in 0 Dimensions

DoS in GUEN from fermionic replicas (EK, 2002)

E i

ZEZ0

IV )(exp)0(~)(~ tdtnn

02244 22 N'n'''t'' IVIVIVIVIVIV

+ boundary conditions

Rn

How it looks in the very end…

Exact Integrability in 0 Dimensions

DoS-DoS in GUEN from fermionic replicas (EK, 2002)

)()2()2(log)(~log 321

2 nEEnnin OZ

042 n'''t't''t VVVVVVV

i

nn ein

G sin21)(~1

lim)( 2220 Z

+ boundary conditions

Rn

22/)(

exp)0(~)(~

0

2V

t

ntntdti nn ZZ

How it looks in the very end…

Exact Integrability in 0 Dimensions

DoS in chGUE from fermionic replicas (EK, 2002)

)()()()()(log)(~log 211

0

ntItKtItKtdtsnss

n OZ

0))(14(2222 't'''n''t IIIIIIIIIIIIIIIIII

)()()(2

)()( 112 sJsJsJ

sss

+ boundary conditions

Rn

2

0

4/2/)()(exp)(~

s

t

tntdtss IIIn

n

Z

Towards Exact Integrability of Replica Field

Theories in 0 Dimensions

Conclusions

Why Replicas? What Are Replicas and The Replica Limit?

Two Pitfalls: Analytic Continuation and (Un) controlled Approximations

Outline

Nonperturbative Methods in Physics of Disorder

Established:

Conclusions

Shown:

Derived:

Link between fermionic replica field theories in 0D and the theory of nonlinear integrable hierarchies Toda hierarchy of replica partition functions intimately related to the unitary symmetry of the replica field theory Exact nonperturbative correlation functions in the GUE, Ginibre’s ensemble of complex random matrices, and in chiral GUE (Dustermaat-Heckman theorem is violated) – by reduction of Toda Lattices to Painlevé equations combined with a “simple minded” analytic continuation

symmetry classes (Pfaff Lattices)

41 ,

Puzzle of bosonic replicas

Questions left unanswered

What is about the uniqueness of analytic continuation?

Is there exact integrability of 0D replica field theories

with interaction?

Workshop on Random Matrix Theory: Condensed Matter, Statistical Physics, and Combinatorics The Abdus Salam International Centre for Theoretical Physics, Trieste, June 29, 2004

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